How to design four-bar function-cognates
Citation for published version (APA):Dijksman, E. A. (1975). How to design four-bar function-cognates. In Fourth world congress on the theory of machines and mechanisms : papers read at the congress at the University of Newcastle on Tyne, September 1975 (pp. 847-853). Mechanical Engineering Publications.
Document status and date: Published: 01/01/1975
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, ,
"~'.. <
.. OW TO DESIGN FOUR-BAR
FUNCTION-COGNATES
E. A. DIJKSMAN
Liverpool Polytechnic. V.K.
SYNOPSIS
The opposite angles
JA
and ~ in a four-bar linkaqe are related by a funetional relationshLp}J. - )J. (t/)l which is produced by variation of the motion variabie
(J.
It will be shown thatan infinite number of four-bars eKistt that produce the same functional relationship.
INTllODUCTION
1. A designer of meehanisms, generally, is on, the look-out for alternative solutions to his problems. If, for instanee, many alternative mechanisms are available, he is able to choose
the best one that will also suit additional requirements such as, for example, those that concern the space that is going to be occupied by the required mechanism.
2. Apart from mechanisms that fundamentally have a different structure or a different kinematic chain, there are those that have the same kinematie ehain, but have different(main) dimensions. Alternative meehanisms of the last type meeting the same initia 1 requirements, are called eognate.meehanisms.
3. In case the mechanisms coneerned are restrieted to linkage mechanisms, we are dealing with cognate linkages. For the designer, it is important to know how to obtain a cognate linkage one from the other. Onee he knows this, he is able to optimize his obtained set of cognates in order to meet additional requirements sueh as the one mentioned above.
4. Depending on what kind of requirements are given initially, different kinds of cognate linkages exist. Mainly, we distinguish between
1. Curve cognates 2. Timed-curve eognates 3. Coupler-cognates, 4. Timed-coupler cognates 5. Function-cognates. © I.Mcc:h.E. 1975
These are the most important ones; but of course the list of types could be extended in length as weIl as subdivided.
Briefly, each type will be explained, but here it is espedally the last type in I.'hich \Je are interested;
5. Ad.l Curve-eognates are those that generate the same (coupier) curve by a coupler-point attached to a moving body of the mechanism. Naturally, the moving body to whieh the eoupler-point is attached, has to Oe the same link in the kinema tic chain that represents all
eurve-eognates.
6. For the four-bar meehanism, for instance, three curve-cognates exist, as is well-known from Roberts' Law [I). '
7. Ad.2 Mostly, mechanisms are driven by a regularly rotating crank. If this erank is a singular bar rotating about a fixed pivot, the crank may be observed as the arm of a clock, that only runs a bit faster than usual. The crank, mostly cal led the input-crank, therefore moves through positions that are governed by a single time-variabie
cp ,
which is called theinput-angl~ or sometimes the motion-variable of the mecham.sm.
8. Since we confine ourselves to mechanisms with one degree of freedom in motion only, a position of a coup Ier-point tracing a coupler-curve, would be solely dependent on a single
variabie which could be the motion variabie of the mechanism that produces the curve. This dependency is governed by a functional relatiou-ship between the position-coordinates of the tracing point and the motion-variable, that represents the time. Therefore, curve-cognates that additionally show the same functional relationship between coupler-point-position and the position of the input-crank, are called
timed-~ cognates.
9. Considering Roberts' Configuration again that demoustrates Roberts' Law, it is clear that only two timed-curve cognates exist in case the four-bar is considered. This is true, because in this configuration there are each time only two cranks that rotate through identical angular displacements.
10. Ad.3 Coupler-cognates are alternative mechanisms with the same kinematic chain and a
~ coupler-plane as weIl as a common frame. For four-bat mechanisms, such cognates do not exist, but they do exist for six-bar linkages of Watts' and Stephenson's form. [2,3,4 and
SJ.
11. Ad.4 With respect to coupler-cognates, timed couplëX=ëognates additionally have the same ---functional relationship between the position co-ordinates of the plane and the position of the input-crank. Naturally, this set of cognates is only a sub-set derived from the coupler-cognates. But, generally, they comprise probably the most important group as far as industrial application is concerned. This caused by the fact that in industry many mechanisms have to be designed bringing objects from one place to the other with a given interval of time.12. Ad.S Function-cognates are cognate
mechan-isms that produce the aame functional relation-ship between the in- and output angle.
13. For a four-bar, generally, they do not exist if the output-angle is the angle between the output-crank and the fixed link. (Again they do exist for six-bar linkages CsJ). However, if we consider the opposite angles ~ and ~, instead of the angles that are measured from the fixed link of the four-bar, it seems that an infinite number of such cognates exist.
14. In this paper, we will confine ourselves to this particular type of cognate only. We will call them four-bar function cognates of the diagonal type, because the opposite angles in the four-bar are the ones that produce the functional relationship in this case.
15. Practical examples of application where the opposite angles and their functional relationship play an important rOle in the design of machines or mechanisma exist. For instanee, machines that are devised for cutting material, are known to rely on this particular kind of functioual relationship. The cutting is then realized by edges which are respectively attached to the coup Ier and to the rocker of a crank-and-rocker mechanism.
848
..
16. It is known also that shutting devices for optical instruments or cameras (diaphragms) are known to rely on the same functional relation-ship.[6]
17. Finally, diagrams th at show the diagonal functional relationship as they actually occur for four-bars, exist. They have been mDde by K.Hain[7] and seem to meet a certain demand from industry.
IS. Clearly. very few people are aware that there are indeed infinite four-bar function cognates of this type. Hbwever. the fund-amentals of such an occurrence have been known EsJ already even in the 19th century. Here, in the paper, we will develop a particular and relatively simple way of obtaining the oue cognate from the other. 8y doing this, we indeed will have devised a short cut in theory that certainly will be of a tremendous advan-tage to the designer and will, hopefully, in this century, open up the knowledge about this subject.
19. Since the derivation of the function
cognate plays an important part for the designer, it is necessary to prove the proposition which will be revealed here.
20. Basically, we will make an extensive use of overconstrained linkages. This is becsuse functiou cognates coqld be mechanically
connected to one another and are then forming a part of an overconstrained linkage.
21. The overconstrained linkage we have in mind here, is known as Kempe's (overconstrained) linkage,[S] sometimes cal led Burmester's focsl mechanism.[9] The reason why it is called a focal mechanism is because the quadruple joint that is connected to the four sides of a four-bar, coincides with one of the two foei of a conic section, that is inscribed in the four-bar quadrilateral.
22. Since a conic section is completely deter-mined by five tangents, an infinite number of conic sections exists that could be inscribed in the four-bar. So, there must be a locus for those foei that in turn could be connected to the four sides. This locus is introduced by
Burmester [9J as the focal curve of the four-bar. (ODe may prove thst it is identical to the
~-point-~ if the four-bar is observed as an opposite pole quadrilateral).
23. It will be shown later that this infinite number of foei that are indeed svailable to the designer, is directly linked up with the exist-ence of an infinite number of four-bar function cognates of the diagonal type.
24. Kempe, who actually found the linkage, has derived it in an unknown manner using
complicated algebraic computations to prove its movability.
' , ' .
25. Burmester, who explored its properties, bas ·found geometrie relationships that cleared up
same of the mysteries tbat hang around the link-age.
26. Finally, Wunderlich [lOJ devised another proof of its movability based on complex numbers.
27. None of them, however, noted a direct connection that exists between the focal linksge and a more trivial type of overconstrained
link-age. which is introduced by either Reuleaux or by Burmester.
28. Because of its advantage of better under-standing, this connection will be explained in some detail here and simultaneously used as a purely geometrie derivation of the focal linksge. Su eh a derivation will keep the number of used formulas to a bare minimum. It simultaneously provides the designer with the shortest proof possible, whereas the geometrie treatment and better understanding might easily lead to new devices and ,other ideas.
DERIVATION OF BURMESTER'S OVERCONSTRAINED LINKAGE [U]
29. If we multiply a four-bar AoABBo about A by a factor, say AS/AB, we obtain a similar four-bar PASF, that may be connected to the first one. (See figure lB). In fact, we now have adjoined the initial four-bar by a dyad SFP.
30. As link SF moves parallel to BBo, it is possible to form the linksge parallelogram SFRB as shown in the figure. Similarly, the link-age parallelogram AoPFQ comes into being. Totally, we now have obtained a linkage which is overconstrained, because it does not comply with GrUbler's general criterium of movability.
31. Since its derivation is relatively simple, we consider this as a more or less trivial case. One may observe, however, that the configuration contains three permanent similar four-bar chains and two linkage parallelograms. Also, the quadruple joint F that is connected to the four sides of the outside four-bar, permanently stays on the diagonal of this four-bar.
32. Finally, one may remark that the linkage constitutes two connected plagiographs of Sylvester.
33. In the next section it will be shown that the linksge so assembied is to be directly linked up with or cognated to Kempe's over-constrained linkage. For this reason the linkage will be called Burmester's cognated configuration as is shown in figure 1.
DERIVATION OF KEMPE'S OVERCONSTRAINED UNKAGE [12]
34. For this, we again start with a given but arbitrary four-bar AoABBo (See the figures 1 and 2). We then choose a symmetrical axis in
© I.Mech.E. 1975
which the four-bar has to be reflected. The symmetrical axis intersects the diagonal ABo of the four-bar at the point F which is to be the focal point later on.
35. We further interseet the side A'A~ of the reflected four-bar and the side BoB at the point
D. Similarly, we interseet A~B~ and AB at the point E.
36. We then rotate the reflected four-bar as a yhole about F until D, E and F join ~
straight-line. As a consequence then the symmetrical axis will also be rotated about F.
37. Therefore, having chosen the point F on a diagonal, say ABo' the symmetrical axis joining
F, could be found such that the three points D, E and F are aligned. Suppose this condition
is attained; we then have two symmetrical four-bars, AoABBo and A'A'B'B', of which the'inter-secting points D, ~ and ~ join a straight-line. 38. We further complete each of the four-bars with four bars connecting F with the sides of each four-bar. (See figure 3). Each of them so constitute a Burmester configuration, that conneets two plagiographs of Sylvester. (See also figure 1). So, we have two configurations of Burmester which are symmetrical with respect to one another.
39. We now conneet them through the turning-joints F and D, and so obtain a many-fold overconstrained singular mechanism but with only one degree of freedom in motion. (See again figure 3).
40. We further recognize the four-bar FROP'. Since F, E and D join a straight-line, we may dilatate or multiply the four-bar geometrically, until the similar four-bar EBDA~ is obtained. !bus, because of this, we may now connect AB and
A~B~ through turning-joint E. (See figure 4).
41. For' abbreviation's sake we will now omit the proof that the so assembied configuration stays symmetrieal. So, assuming that symmetry is not destroyed throughout the motion, we may similarly conneet the two symmetrical config-urations at the points E' and D'. (See again figure 4).
42. Finally then, we recognize a sub-chain which is the four-bar E'A'DBo and a point F that is connected to the sides through the bars FS'. FP', FR and FQ respectively. (Sinee F joins the diagonal ABo and the assembied configuration is a symmetriealone, we can see that the opposite sides E'A' and BoD subtend angles with sum ~ at
F. Further, 1 S'A'F
=
1 FAS=
~ BoFR and so on).43. We so obtain the focal linkage.
figure lA). (See
44. This method of derivation actually conneets Burmester's overconstrained linkage with the focal linkage of Kempe-Burmester.
45. Both mechanisms, if campared, have a focal point which is eonnected to tbe sides of tbe four-bar. In BuEmester's configuration bowever, we recognize two.opposite parallelograma. In
the focal linksge tbere is none. Also, in BUEmester's configuration we bave three direetly similar four-bars. In tbe foeal linksge we have only two pairs of reflected similar four-bars.
46. Indeed, A.B.Kempe [S], wbo devised tbe focal linkage in the first place, found each time two opposite four-bars being reflected similar to one anotber, tbat are CODtained in tbe linkage. It is attained directly by choosing a focal point F within the outside q~rangle, suchthat opposite sides subtend augles with sum w at F. If F is cbo.en outside the quadrangle E'A'DBo' the angles subtended must be equal.
47. The locus of points F comprising such a property is the focal curve mentioned earlier. Aay point F on this cu~o meets the condition and may be taken as focal point that is to be cànnected to the sides. Especially so, since such a point allows us to form opposite quad-rangles within the four-bar that are reflected similar to eacb otber. This is carried out by drawing the transversal lines that COnnect F with tIle joints of the outside four-bar. We then recognize four pairs of triangles that bave to be reflected similar to each other. This property enables us to adjoin the bars that conneet tbe foeal point to the four sides. DESIGN OF KEMPE'S FOCAL LINKAGE
4S. Although it would be possible to design the linkage using its focal properties. the actual design of the linkage would be much simpler i f
we are able to avoid using the focal curve. Therefore, we will introduce another way to design the linkage. For this it is necessary to derive some properties of the focal linkage. 49. Fram figure 4, for instanee, we derive that
A'P' AP ~ P'F - PF • QBo Rence, P'F.FQ • A'P'.QBo Similarly, Therefore, P'F.FQ a E'Q.P'D
And so, from (1) and (2), we find that A'P'
pïi)
(See also the figures 1 and 5) In the same way, we Hnd that
S50 (1) (2) (3) S'F.FR
•
S'A'.BoR (4) S'F.FR•
E'S' .IID (5)and so, equally
E'S' BoR (6) S'A'
• iD
50. Now, we start the design with the choice
of the outside four-bar that isO E'A'DBo (See figure 5). Further, to avoid using the focal curve, we choose the point Q on the line E'Bo instead. The problem then is to find the coordinated focal point F and to Hnd the turn-ing-joints P', S' and R.
51. To make it easier to solve this problem, we
change the motion variable of the outside four-bar until A'D comes parallel tÖ E'BQ• Then,
since ~ A'P'F c ~ FQBo' QF and FP' loin.the same
line in the design position of the linkage.
52. Also, according to eq.(3), the line QP' has to join then the intersection point r of the uprising sides E'A' and BoD.
Thus, r . (E'A',BoD) and P' c (Qr,AID).
Bence, we now have established the sum (P'F + FQ).
53. Through eq.(l) it is additionally possible to determine the value of the product (P'F.FQ). Consequently, we are then able to calculate or to construct the lengths P'F and FQ separately. 54. This is carried out through aquadratic equation of which there are two roots. (Sinee the quadratic equation may have imaginary or complex roots, it is clear that not for all points Q on E'B real solutions exist. If such is tbe case it gecomes necessary to relocate the point Q until real solutions are available).
55. Assuming therefore, that the roots are real, we find two focal points Fl and F2, either of whieh could be connected to the points P' and Q.
(Clearly, the quadrangle QFIP'F2 must resemble a contra-parallelogram).
56. Suppose we now further choose one of the possible focal points F. We then find R, using the fact that ~ E'A'F
=
~ RFBo' Finally, we establish the point S' by drawing RS' parallel to A'D and EIBo' '57. We note that ~ A'S'F • ~ FRBo' Therefore,
IJ S'FRr'is a cyclic quadrangle. Thus, at r the tangent p to the circle circumscribed about the quadrangle, meets the condition that
~ prD • ~ rS'R • ~ rA'D.
DERIVArION OF THE FOUR-BAR FUNCTION-COGNATES
SS. Suppose we want to find one of the infinite four-bar function-eognates that exist here. We further assume that the, initial four-bar we start the design with is represented by the four-bar E'S'FQ. (See figure 6). Still one degree of freedom in design is left to us. Therefore, we may use this in order to simplify matters.
For instance~ if we take ~ (A'D,E'Bo) • 0, it
then follows that A'D//S'R//E'Bo and also that f, P', F and Q join the same line. This is in accordance with our earlier findings in this particular position. Later, we will change the initial design position of 0 E' S 'FQ through the lI:lCtion variable
c.p,
then repeat the procedure, and so obtain an infinite.number of solutions all different from each other.59. For a given motion"Variable the four-bar function-cognate E'A'DBo just described, is then to be found as follows:
a. Interseet E'S' and QF at the point
r
b. Draw a circle joining the points f, S'and F
c. Next, draw the tangent p to the circle at f d. Further, make ~ pfBo
=
~ fE'Q • ~ and drawthe line rBa.
e. Draw S'R parallel to E'Q
f. Interseet S'R,fBo and the circle at the one point R
g. Interseet rBa and E'Q at the center Ba h. Make l!A'S'F '" àFRBo and so determine
point At
i. Draw A'D parallel to E'Bo
j. Intersect A'D and rBo at the point D k. The function-cognate obtained isOE'A'DBo ' 1. Vary the motion variable ~ and so obtain
other function-cognates producing the same functional relationship J! = \I (lP) for the opposite angles.
60. All solutions obtained in thia way,are four-bar function-cognates from one another. (See figures 7A and 7B).
CONCLUSION
61. It is proved here that the functional relationship between oppoaite angles in a four-bar linkage are produced by an infinite number of these1inkages.
62. Further, four-bar linkages generating the same functional relationship \I • J!(~) have been named four-bar function-cognates of the diagonal type. They are derived from one another using Kampe's overconstrained eight-bar linkage that is proved to be cognated to Burmester's over-constrained configuration made up by two plagiographs of Sylvester.
63. The paper also shows a simple, geometric way to design the mentioned cognates directly. 64. The freedom of choice that is finally obtained here,may now be used by the designer, to incorporate such demands that have to do with the available space for the mechanism.
ACKNOWLEDGMEN'rS
The author of the present paper would like to thank the support of the Science Research Council under Grant No. B/RG/1016.5, and also the support provided by the Mechanical Engineering Department of the Liverpool POlytechnic.
REFERENCES [IJ
[sJ
DIJKSMAN, E.A. "How to campase mechanisms with parallel moving bars". De Ingenieur 82 (1970) 47, p.W17l-l76.
DIJKSMAN, E.A •. "Six-bar cognates of Watt's farm". Transactions of the A.S.M.E. Series B. Journalof Engineering for Industry, 93 (1971), p.183-l90. DIJKSMAN, E.A. "Six-bar cognates of a Stephenson Mechanism" • Journal of Mechanisms, 6 (1971) 1, p.3l-57. DIJKSMAN, E.A. "Five-bar curve and coup Ier cognates having two cranks that rotate with the same speed". I.F.T.O.M.M. International Symposium on Linkages and Computer Design Methods. Bucharest, Romania, June 7-13, 1973, Volume A. Paper A18, pp.230-255.
DIJKSMAN, E.A. '~bene sechsgliedrige çetriebe als abgewandelte FUhrungs-und
Funktionsgetr:i:ehe". VDl-Berichte (1973) Essener Getriebe - Tagung 1973. Methodik bei der Auswahl und Konstruktion von
Getrieben. Kinematik. Dynamik. Lebensdauer. KIPER, G. "Die Wah1 des Mechanismus als konstruktive Teilentscheidung".
VDl-Berichte (1973). Getriebetagung 1973; Methodik bei der Auswah1 und Konstruktion von Getrieben. Kinematik, Dynamik, Lebensdauer.
[7J HAIN,K. l."Getriebe - Atlas für verstellbare Schwing-Dreh-Bewegungen". Friedr. Vieweg
& Sohn GmbH, Verlag, Braunschweig (1967).
2. Atlas für Getriebe-konstruktionen. Texteil. Vieweg. Braunschweig (1972).
[8J
KEMPE, A.B. "On conjugate four-pieceliok-ages". Proceedings of the London Mathem-atical Society. 9 (1878). p.133-l47. [9J BURMESTER, L. "Die Brennpunktmechanismen"
Zeitschrift für Mathematik und Physik,
8 (1893), p.193-223.
[lOJ WUNDERLICH, W. "On Burmester's focal mech-anism and Hart's straight-line motion". Journalof Mechanisms. 3 (1968), 2, p.79-86.
[U]
DlJKSMAN. E.A. "How to replace the four-bar coup Ier motion by the coupler motion of a six-bar mechanism that does not con ta in a parallelogram". Romanian Journalof Technical Sciences. Applied Mechanies(M4canique Appliqu4es) (1974).
[12J
DIJKSMAN, E.A. "Kempe's Focal LinkageGeneralized, particularly in connection with Hart's second straight-line mechanism". Mechanism and Machine Theory 1974.
852
E'
Figure lA
Kempe 's Focal Mechonism wilh conslroinl molion
118781
Figure 1 B
Burmeslers' cognoted configurolion
F.gure 3
\,symmetncol ox.s
\
\Overcons trolned Linkoge w.lh one degrce ollreedom in moIIon Äo I I I I I I
Bl..,
I
" I Figure 2 '~ B~ s mmelricolox.sSymmetr.col oxis chosen such thol D.Eond F join 0 slroight-line
p: p' A;,
ji'--r---..,.---liJ..,I
/ / /
I S' I ' \ symmetricol aXIs'\
f.gure 4Two symmelr.col conl.gurotions ol Burmester in cognotlon
\
.. -,
.
I I I I I I I I I I I \ \ \ \ I / / I / \ \ \ \, ,
/ " ";/.,...,....,,,'"
", Ea
- - - ~Ie. rlr'~p
I/ I " /1 1 \ /1 1 \/ I'
\
/ I
I \ / I I \ / I \ \ /~
\
:
/ I I : / f I 1 1 / 1 / lp''0
!
I , , "'/,/ -,'P'F. Fa : A'P'. OB" = E'tl. Figure5
Design Position ol Kempe's Focal Mechanism
© I.Mech.E. 1975 coUineation·oxis E' E' p A ~ S'F -t:. F R Bo oE'S'F Q and oE'~OB. Figure ó
Two alternative tour-bars havlng the same lunclional relationship (.I=(.I!IjlJ
Infinlte Function-Cognates wlth lour-bars
o
o
E' Q 6.
Figure 7A
Funchonal relatIonship through dlametrically opposlle angles In a lour-oor
secondory Itnk
~
'
F Ijl j.. .__
.__
. -E' Q Figure 7 BInhnlte cognote lunchon generators wlth lour-bars
,.'
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